Why does DFT based channel estimation in OFDM fail at non-pilot subcarriers?

Hi,

I want to do DFT based estimation for ODFM. The estimation is working well for the pilot position but it is not working for the unknown positions.

I am attaching the matlab code below.

 

clc

close all

clear all

Nfft = 128;

No_sc = 12;

No_RB = 6;

No_data_sc = No_RB * No_sc;

idx_data_sc = [-36:-1 1:36];

idx_pilot = [-36:6:-1 1:6:36];

data = abs(randn(No_data_sc,1)>0.5)+1;

% idx_data_sc+Nfft/2+1;

% idx_pilot+Nfft/2+1;

% idx_pilot+Nfft/2+1-(Nfft-No_data_sc)/2+3;

mod_carriers = zeros(Nfft,1);

mod_carriers(idx_data_sc+Nfft/2+1,1) = data;

% mod_carriers = fftshift(mod_carriers);

t_data = sqrt(Nfft)*ifft(mod_carriers);

cp_data= [t_data(end-9:end,1); t_data];

h(1) = 1.5;

h(2) = 1;

r_data = conv(h,cp_data);

r_data = r_data(1:length(cp_data),1);

% r_data = cp_data;

de_cp = r_data(11:end,1);

de_fft = fft(de_cp)/sqrt(Nfft);

de_fft = fftshift(de_fft);

f_data = de_fft(idx_data_sc+Nfft/2+1,1);

H = fft(h,Nfft);

H = H.';

H = H(idx_data_sc+Nfft/2+1,1);

f_data = f_data./conj(H);

H_p = H(idx_pilot+Nfft/2+1-(Nfft-No_data_sc)/2+3);

H_p = [ H_p(1:6,1);0;H_p(7:12,1)];

h_p = ifft(H_p);

H_r = fft(h_p, No_data_sc+1);

plot(1:72,abs(H.^2));

hold on

plot(1:73,abs(H_r.^2), 'o');

 

 

%%%%%%%%%%%%%%%%%%%%%%%%

figure;

fsMHz = 1.4;

[Pxx,W] = pwelch(cp_data,[],[],128,1.4);

plot([-64:63]*fsMHz/128,10*log10((Pxx)));

xlabel('frequency, MHz')

ylabel('power spectral density')

title('Transmit spectrum OFDM (based on 802.11a)');